Theorem dcomex 10453

Description: The Axiom of Dependent Choice implies Infinity, the way we have stated it. Thus, we have Inf+AC implies DC and DC implies Inf, but AC does not imply Inf. (Contributed by Mario Carneiro, 25-Jan-2013.)

Assertion

Ref	Expression
dcomex	⊢ ω ∈ V

Proof of Theorem dcomex

Dummy variables 𝑡 𝑠 𝑥 𝑓 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		1n0 8478	. . . . . . 7 ⊢ 1o ≠ ∅
2		df-br 5108	. . . . . . . 8 ⊢ ((𝑓‘𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛) ↔ ⟨(𝑓‘𝑛), (𝑓‘suc 𝑛)⟩ ∈ {⟨1o, 1o⟩})
3		elsni 4604	. . . . . . . . 9 ⊢ (⟨(𝑓‘𝑛), (𝑓‘suc 𝑛)⟩ ∈ {⟨1o, 1o⟩} → ⟨(𝑓‘𝑛), (𝑓‘suc 𝑛)⟩ = ⟨1o, 1o⟩)
4		fvex 6895	. . . . . . . . . 10 ⊢ (𝑓‘𝑛) ∈ V
5		fvex 6895	. . . . . . . . . 10 ⊢ (𝑓‘suc 𝑛) ∈ V
6	4, 5	opth1 5455	. . . . . . . . 9 ⊢ (⟨(𝑓‘𝑛), (𝑓‘suc 𝑛)⟩ = ⟨1o, 1o⟩ → (𝑓‘𝑛) = 1o)
7	3, 6	syl 18	. . . . . . . 8 ⊢ (⟨(𝑓‘𝑛), (𝑓‘suc 𝑛)⟩ ∈ {⟨1o, 1o⟩} → (𝑓‘𝑛) = 1o)
8	2, 7	sylbi 220	. . . . . . 7 ⊢ ((𝑓‘𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛) → (𝑓‘𝑛) = 1o)
9		tz6.12i 6908	. . . . . . 7 ⊢ (1o ≠ ∅ → ((𝑓‘𝑛) = 1o → 𝑛𝑓1o))
10	1, 8, 9	mpsyl 69	. . . . . 6 ⊢ ((𝑓‘𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛) → 𝑛𝑓1o)
11		vex 3457	. . . . . . 7 ⊢ 𝑛 ∈ V
12		1oex 8469	. . . . . . 7 ⊢ 1o ∈ V
13	11, 12	breldm 5896	. . . . . 6 ⊢ (𝑛𝑓1o → 𝑛 ∈ dom 𝑓)
14	10, 13	syl 18	. . . . 5 ⊢ ((𝑓‘𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛) → 𝑛 ∈ dom 𝑓)
15	14	ralimi 3101	. . . 4 ⊢ (∀𝑛 ∈ ω (𝑓‘𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛) → ∀𝑛 ∈ ω 𝑛 ∈ dom 𝑓)
16		dfss3 3923	. . . 4 ⊢ (ω ⊆ dom 𝑓 ↔ ∀𝑛 ∈ ω 𝑛 ∈ dom 𝑓)
17	15, 16	sylibr 237	. . 3 ⊢ (∀𝑛 ∈ ω (𝑓‘𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛) → ω ⊆ dom 𝑓)
18		vex 3457	. . . . 5 ⊢ 𝑓 ∈ V
19	18	dmex 7910	. . . 4 ⊢ dom 𝑓 ∈ V
20	19	ssex 5289	. . 3 ⊢ (ω ⊆ dom 𝑓 → ω ∈ V)
21	17, 20	syl 18	. 2 ⊢ (∀𝑛 ∈ ω (𝑓‘𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛) → ω ∈ V)
22		snex 5408	. . 3 ⊢ {⟨1o, 1o⟩} ∈ V
23	12, 12	fvsn 7183	. . . . . . . 8 ⊢ ({⟨1o, 1o⟩}‘1o) = 1o
24	12, 12	funsn 6590	. . . . . . . . 9 ⊢ Fun {⟨1o, 1o⟩}
25	12	snid 4626	. . . . . . . . . 10 ⊢ 1o ∈ {1o}
26	12	dmsnop 6216	. . . . . . . . . 10 ⊢ dom {⟨1o, 1o⟩} = {1o}
27	25, 26	eleqtrri 2861	. . . . . . . . 9 ⊢ 1o ∈ dom {⟨1o, 1o⟩}
28		funbrfvb 6935	. . . . . . . . 9 ⊢ ((Fun {⟨1o, 1o⟩} ∧ 1o ∈ dom {⟨1o, 1o⟩}) → (({⟨1o, 1o⟩}‘1o) = 1o ↔ 1o{⟨1o, 1o⟩}1o))
29	24, 27, 28	mp2an 705	. . . . . . . 8 ⊢ (({⟨1o, 1o⟩}‘1o) = 1o ↔ 1o{⟨1o, 1o⟩}1o)
30	23, 29	mpbi 233	. . . . . . 7 ⊢ 1o{⟨1o, 1o⟩}1o
31		breq12 5112	. . . . . . . 8 ⊢ ((𝑠 = 1o ∧ 𝑡 = 1o) → (𝑠{⟨1o, 1o⟩}𝑡 ↔ 1o{⟨1o, 1o⟩}1o))
32	12, 12, 31	spc2ev 3564	. . . . . . 7 ⊢ (1o{⟨1o, 1o⟩}1o → ∃𝑠∃𝑡 𝑠{⟨1o, 1o⟩}𝑡)
33	30, 32	ax-mp 5	. . . . . 6 ⊢ ∃𝑠∃𝑡 𝑠{⟨1o, 1o⟩}𝑡
34		breq 5109	. . . . . . 7 ⊢ (𝑥 = {⟨1o, 1o⟩} → (𝑠𝑥𝑡 ↔ 𝑠{⟨1o, 1o⟩}𝑡))
35	34	2exbidv 1957	. . . . . 6 ⊢ (𝑥 = {⟨1o, 1o⟩} → (∃𝑠∃𝑡 𝑠𝑥𝑡 ↔ ∃𝑠∃𝑡 𝑠{⟨1o, 1o⟩}𝑡))
36	33, 35	mpbiri 261	. . . . 5 ⊢ (𝑥 = {⟨1o, 1o⟩} → ∃𝑠∃𝑡 𝑠𝑥𝑡)
37		ssid 3956	. . . . . . 7 ⊢ {1o} ⊆ {1o}
38	12	rnsnop 6224	. . . . . . 7 ⊢ ran {⟨1o, 1o⟩} = {1o}
39	37, 38, 26	3sstr4i 3985	. . . . . 6 ⊢ ran {⟨1o, 1o⟩} ⊆ dom {⟨1o, 1o⟩}
40		rneq 5924	. . . . . . 7 ⊢ (𝑥 = {⟨1o, 1o⟩} → ran 𝑥 = ran {⟨1o, 1o⟩})
41		dmeq 5891	. . . . . . 7 ⊢ (𝑥 = {⟨1o, 1o⟩} → dom 𝑥 = dom {⟨1o, 1o⟩})
42	40, 41	sseq12d 3967	. . . . . 6 ⊢ (𝑥 = {⟨1o, 1o⟩} → (ran 𝑥 ⊆ dom 𝑥 ↔ ran {⟨1o, 1o⟩} ⊆ dom {⟨1o, 1o⟩}))
43	39, 42	mpbiri 261	. . . . 5 ⊢ (𝑥 = {⟨1o, 1o⟩} → ran 𝑥 ⊆ dom 𝑥)
44		pm5.5 364	. . . . 5 ⊢ ((∃𝑠∃𝑡 𝑠𝑥𝑡 ∧ ran 𝑥 ⊆ dom 𝑥) → (((∃𝑠∃𝑡 𝑠𝑥𝑡 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛)) ↔ ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛)))
45	36, 43, 44	syl2anc 596	. . . 4 ⊢ (𝑥 = {⟨1o, 1o⟩} → (((∃𝑠∃𝑡 𝑠𝑥𝑡 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛)) ↔ ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛)))
46		breq 5109	. . . . . 6 ⊢ (𝑥 = {⟨1o, 1o⟩} → ((𝑓‘𝑛)𝑥(𝑓‘suc 𝑛) ↔ (𝑓‘𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛)))
47	46	ralbidv 3187	. . . . 5 ⊢ (𝑥 = {⟨1o, 1o⟩} → (∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛) ↔ ∀𝑛 ∈ ω (𝑓‘𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛)))
48	47	exbidv 1954	. . . 4 ⊢ (𝑥 = {⟨1o, 1o⟩} → (∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛) ↔ ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛)))
49	45, 48	bitrd 282	. . 3 ⊢ (𝑥 = {⟨1o, 1o⟩} → (((∃𝑠∃𝑡 𝑠𝑥𝑡 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛)) ↔ ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛)))
50		ax-dc 10452	. . 3 ⊢ ((∃𝑠∃𝑡 𝑠𝑥𝑡 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛))
51	22, 49, 50	vtocl 3523	. 2 ⊢ ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛)
52	21, 51	exlimiiv 1964	1 ⊢ ω ∈ V

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145   ≠ wne 2957  ∀wral 3078  Vcvv 3453   ⊆ wss 3902  ∅c0 4282  {csn 4587  ⟨cop 4593   class class class wbr 5107  dom cdm 5659  ran crn 5660  suc csuc 6363  Fun wfun 6531  ‘cfv 6537  ωcom 7866  1_oc1o 8452

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402  ax-un 7740  ax-dc 10452

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-fv 6545  df-1o 8459

This theorem is used by:  axdc2lem  10454  axdc3lem  10456  axdc4lem  10461  axcclem  10463  precsexlem10  28489  seqsex  28558  noseqex  28562